AbstractThe z-distribution of pulsars in the vicinity of the Sun
is investigated using data from the ATNF pulsar catalogue and the recent
model for the Galactic distribution of free electrons (NE2001). It
is found that the z-distribution of pulsars with L>10 mJy kpc2is exponential with a characteristic scale height of about 350 pc. Evidence
of pulsar oscillations in the direction perpendicular
to the Galactic plane is presented.

It is accepted that the z-distribution of the Galactic
Population I objects is approximately exponential (Binney & Merrifield 1998).
This kind of the observed distribution can be explained by the
dynamic equilibrium that exists in the Galaxy.

It is known that the Galactic gravitational potential causes
oscillation of the Galactic Population I objects in the direction
perpendicular to the Galactic plane. If Population I subsystem objects have
a Gaussian
velocity distribution in the Galactic
plane and if the subsystem is "well-mixed'', the z-distribution
of the objects is approximately exponential (Oort 1965). The
scale height of the z-distribution of pulsar progenitors is of
the order 50-100 pc. On the other hand according to relatively
recent data the z-distribution of pulsars is approximately an exponential
with an estimated scale height of 600 pc (Lyne & Graham Smith 1998).

Today this discrepancy is explained by a suggestion that pulsars are
runaway stars (Gunn & Ostriker 1970). Gunn & Ostriker (1970) analyzed statistically the
data of 41 pulsars known at that time and found that the scale height
in the z-distribution was about 120 pc, similar to that of A stars.
The picture of the Galactic distribution of the pulsars depends strongly
on the model of the free electron distribution in the Galaxy. Different
models give different z-distributions. Gunn & Ostriker (1970) adopted a hypothesis
according to which the pulsar luminosity should decay exponentially, with a
time scale of about 4.55 Myr. As the pulsars are born near the Galactic plane as
runaway objects, they should have moved some distance away from the
Galactic plane before their luminosities fall below
and
they effectively "die'', i.e. they become unobservable. The
combination of the pulsar motion away from the Galactic plane with
a decay of the luminosity should determine the observed
z-distribution. Therefore, although pulsars and Population I
subsystems show a similar exponential z-distribution, the
mechanisms of the formation of the z-distribution should differ.

In the present paper we have used data from the ATNF pulsar
catalogue version 1.15 (www.atnf.csiro.au/research/pulsar/psrcat/) and the recent model for the
Galactic distribution of free electrons (NE2001) by Cordes & Lazio (2002)
to estimate the distances and their uncertainties. While
studying the z-distribution of pulsars we have used pulsars
located inside a cylinder with 3 kpc radius centered on the
Sun, and with the axis along the z-direction. We discuss the
problem of possible oscillation of the pulsars in the
direction perpendicular to the Galactic plane.

While studying the z-distribution of pulsars one should take
into account various systematic errors caused by (1) the
observational biases of the surveys; (2) instrumental selection
effects (Lyne et al. 1985); and (3) random and systematic errors caused
by the used distance model (Gunn & Ostriker 1970). Detailed analysis
of these biases, which is beyond the scope of this paper, needs
Monte Carlo simulations of a synthetic population of pulsars.

Nowadays, it is accepted that from most present pulsar surveys it is
not possible to make reliable statements about the population
of pulsars with luminosities lower than about 10 mJy kpc2(Lyne et al. 1998; Lorimer et al. 1993).
Therefore, we limited ourselves to the sample of pulsars with L> 10 mJy kpc2 located inside a cylinder with 3 kpc radius centered on the Sun.

It is also accepted that the pulsar distance scale is
statistically consistent with other methods of deriving
galactic distances and that it has no systematic errors. Random errors
in the distance measurements affect the estimation of the scale height
in the z-distribution. According to Gunn & Ostriker (1970) for a logarithmic-normal
distribution law of the observed distances
the scale height of the observed z-distribution is related to
the real one as

(1)

For a distance uncertainty
the
multiplier
and can be neglected
in the following analysis. Note that the distances to the
OB stars (the pulsar progenitors) are measured with the same accuracy.
We also used the method of testing hypotheses (student's t-test) while comparing
two mean values from different samples (Devore 1995).

According to the new distance model there are 377 pulsars within 3 kpc cylindrical radius from the Sun, excluding the millisecond
pulsars and the pulsars that are members of the globular clusters. As
mentioned above a reliable statistical analysis can be made for a
luminosity-limited sample with L> 10 mJy kpc2, where
L=S400r2; S400 is the flux (in mJy) at 400 MHz and r is the distance in kpc. With this restriction our sample
comprises 234 pulsars. To filter the data we
use the estimate of the distance uncertainty .
To estimate
the average 16th and 84th percentiles the dithered
distance estimates are used. They define a nominal 68% confidence
range in the model distance for each pulsar. These percentiles are
given in NE2001 (Lazio 2002).

First let us consider the dependence of the observed (b is a Galactic latitude) on the distance uncertainty .
Figure 1 shows the relation between z and .
The
uncertainty increases with increasing z and for a higher zthe uncertainty greatly exceeds the criterium
.
It
should be mentioned that according to the new distance model the
number of pulsars with
is not large; it comprises
about 10% of the sample. Some pulsars (about 3% of the
sample) located near the Galactic plane also have large
uncertainties which must be caused by their location near the
regions of irregularities in the free electron distribution.

Let us now consider the distribution of the absolute value of z for the sample with accurate distances (
). Figure 2 (the right panel) shows that the distribution decays
roughly exponentially, with scale
height parsecs and a median value of about 270 pc. The
whole sample is located inside a layer with a thickness of about 2 kpc from the Galactic plane. The z-distribution of the whole sample
can also be approximated by an exponentially decaying function
(left panel in Fig. 2) with scale height pc and a
median value of about 285 pc. From these results we can argue that the scale
height of the observed z-distribution is about 350 pc which is less than
the accepted value of 600 pc (Lyne & Graham Smith 1998).

Figure 2:
Histograms of the |z|-distribution of the
pulsars for the whole sample ( left panel) and the sample with
( right panel).

It is interesting to consider the characteristic age distribution
of the pulsars from our sample of N=234 pulsars. To
take advantage of the law of large numbers while comparing the mean
values of the parameters, we have arranged the sample according to
increasing age and then divided it into 8 subgroups, consisting of ni=30 pulsars (i=1...7), except for the last subgroup
(n8=24). The characteristic age varies in a wide range,
from about 1000 year (the Crab pulsar) up to 600 Myr
(J1834-0010). In Fig. 3 we show dependance of the
relative frequency of pulsars per age interval for the
subgroups
vs.
,
where
is the age interval and
is the mean
age for the ith subgroup. One effect that may be able to alter
the observed distribution is the instrumental bias
against the detection of short period pulsars. These missing
pulsars are likely to be young, luminous objects, staying near the Galactic
plane and therefore
particularly prone to the effects of pulse broadening in the
interstellar medium (Lyne et al. 1985). Therefore, the observed value of
for young pulsars will be depressed relative to its unbiased
value for the underlying population. So the age distribution for
pulsars younger than 10 Myr seems to show an exponential decay.

Figure 3:
The characteristic age distributions. The
error bars represent the standard deviation for
and
for the corresponding subgroup.

Let us consider a dependance of the mean luminosity
(
L=S400r2) on
for the sample. The dependance for the
8 age intervals is given in the left panel of Fig. 4. The
error bars mainly overlap, so we need to use
methods of testing these hypotheses. To compare the mean
values
of two different (ith and i+1th) subgroups we
use the following statistics

(2)

where
is the mean value,
Si,i+1is the standard deviation and
ni,i+1 is the number of pulsars
in the corresponding subgroups. If both subgroups have the same
expected means and ni and ni+1 both are large numbers, then
according to the central limit theorem and the
law of large numbers it can be shown, that
Zni,nj has an
approximately standard normal distribution (Devore 1995).
As a null hypothesis (H0) we
accept that two expected means are equal and as for
the alternative (H1) hypotheses
we use the inequality between the computed means. We chose a 90% confidence level. If the null hypothesis is accepted for two
subgroups (i and i+1) we unite the subgroups and repeat the
procedure from i=1. In the opposite case,
when the null hypothesis is rejected, we accept the alternative
hypothesis and continue the procedure for the second and the
successive subgroups (i+1 and i+2) and so on.
The results of the described procedure for the mean luminosity
is shown in the right panel of Fig. 4. As one can see from
the right panel of Fig. 4, the dependance of the mean
luminosity on
does not show the expected exponential
decay.

Figure 4:
The mean luminosity distribution, before
( left panel) and after ( right panel) using of the testing hypotheses method.

We used the same treatment for the dependance of
vs. for the sample. The results are shown in Fig. 5. It is
clear that
increases with age up to its maximum value
which is reached near the age of 21 Myr, and then a significant reduction
is observed, which is, in our opinion, evidence of oscillation
in z-direction.
One can argue that the obtained evidence of pulsar oscillations
in the z-direction (perpendicular to the Galactic plane) might be
caused by the biases in the distance model used. To avoid this
argument we propose to use a method of revealing the
oscillations that is independent of the distance model.

As known, the observer is located near the Galactic plane.
Therefore, a stellar object moving in the z-direction ()
should show a variation of the value of
in
accordance with its motion relative to the Galactic plane.
The pulsars are apparently born near the Galactic plane with high
peculiar velocities and, on the average, move to higher
z-distances during their lifetime. In this case we should expect
an increase of the average
of the observed pulsars. If
there exists an oscillation it should be seen from the behavior
of
with .
So we applied the method of testing
hypothesis to find the
dependance of
on .
The results are given in
Fig. 6. It is clear that Figs. 5 and 6 show almost the same behavior. Thus we obtain further
evidence of the oscillation of pulsars in the z-direction. It should
be noted that the dependance of both
and
on
shows the same quarter-period of oscillation which is about
21 Myr, near to the nowadays accepted value of 25 Myr for
the Population I objects (Moffat et al. 1998).

There is a number of suggested mechanisms to explain the pulsar age distribution (see e.g. Lyne & Graham Smith 1998, and references
therein). We believe that the simplest possible explanation of such a distribution in statistical terms is the
following. Let us assume that characteristic ages correspond to real ages and that the creation rate of pulsars is
constant during the whole interval of .
Let us also assume that in each stage of pulsar evolution, before
a pulsar reaches the death line in the (
) diagram, there is a non-zero probability that a pulsar becomes
undetectable in a time interval of 1 Myr. Appropriate mechanisms might be: luminosity decay with age (Gunn & Ostriker 1970)
or constant luminosity until an abrupt cut-off presumably due to pulse nulling (Phinney & Blandford 1981), surface magnetic field
evolution due to the plate movement (Gil et al. 2002; Ruderman et al. 1998), a pulsar beaming dependance on spin period which causes older
pulsars to become difficult to detect (Biggs 1990), some unmodelled effects related to the selection criteria in the analysis etc.
If this probability does not depend on the age of pulsars the distribution of pulsars in age can be approximated by an
exponential decay function. The analysis of our sample shows that the observed distribution of pulsars at
characteristic ages can be properly approximated by a third order exponential decay. We have found that the third
order in exponential decay was conditioned due to young pulsars being missed because of pulse broadening, and by the
existing long tail of older pulsars. We have estimated the fraction of missing pulsars according to the method proposed
by Lyne et al. (1985) and found that the approximation of the age distribution can be reduced to a second order
exponential decay. So the characteristic age distribution for our sample indicates that we have a composition of two
exponential distributions with different scale-heights (i.e. with
different "death-rates'').

As is shown in Fig. 4 dependance of
on
does not show the
same exponential decay as the whole sample. But some luminosity decay is observed
in both the first six combined subgroups and the last two combined subgroups. We
cannot make any statement on the character of the decay because for both sets of
combined subgroups we only have two points of data. It seems that this decay is
not exponential as was proposed by Gunn & Ostriker (1970), because in the presence
of the field exponential decay the characteristic age varies non-linearly with real
(chronological) age and the predicted distribution of
must be non-exponential
,
where
(Lyne et al. 1985).

In our opinion the observed luminosity and z-distributions can be explained in terms
of two distinct population of pulsars (see Harrison et al. 1993; Arzoumanian et al. 2002, and references therein);
one born with low velocities and low magnetic fields, and the other
born with high velocities and high magnetic fields.

In this case, if the characteristic ages correspond to true ages we expect that those
pulsars that have velocities high enough to escape the gravitational potential of the Galaxy
will move away so that they soon become unobservable. The remaining part of the sample should
oscillate in the direction perpendicular to the Galactic plane. These pulsars
should gradually become undetectable with age due to one of the mechanisms described above.
Combination of these effects with with the presumably constant
pulsar birth rate in the Galactic plane should give the observed z-distribution.
The maximal separation of pulsars from the Galactic plane is about 2 kpc. But, as is
clear from Fig. 2, the distances to the pulsars that have higher |z| are
estimated with very low accuracy, some of them have
.
Therefore, we have
actually no information about their real location. They may be located much
further and actually be escaping objects.

We think that the determining factor in the revealing of pulsar oscillations must be
a proper motion study of older pulsars.

Acknowledgements

We would like to thank the anonymous referee for the
constructive criticism and useful remarks which helped to
improve the paper. G.M. was supported in part by the Grant 1 P03D 029 26 of the Polish State Committee for Scientific
Research.